We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis. A continuum is said to be complete if the Hessian of the stored energy with respect to the second-gradient variable is locally positive definite, so that every nonzero admissible increment of the placement second gradient is quadratically controlled in the highest-order part of the energy about each configuration. Starting from the Principle of Virtual Work, we derive the constitutive relations, equilibrium equations, and admissible boundary interactions for a broad class of fibrous second-gradient continua whose stored energies depend on fiber stretch, stretch gradients, and curvature. The theory is then applied to several continua motivated by pantographic microstructures. Classical pantographic sheets are shown to be incomplete, while bi-pantographic fabrics enlarge the class of components of the second gradient detected by the energy but remain incomplete. Finally, we formulate a tri-pantographic continuum associated with a proposed three-family architecture and prove that it is complete. The examples illustrate how microstructural architecture influences both the completeness properties of an effective continuum and the pointwise form of its higher-order boundary interactions.

In this communication we propose a discrete Cosserat rod formulation in which a slender elastic rod is represented as a chain of rigid bodies (nodes) coupled by compliant elastic forces and moments acting between adjacent node pairs. Discrete dilatation, shear, torsion and curvature strain measures are evaluated from the relative kinematics of each node pair, while the constitutive behavior is expressed in compliance form through independent stress degrees of freedom. We show that the resulting model arises rigorously from a mixed Petrov–Galerkin Cosserat rod finite element formulation (FEM) at linear kinematic interpolation order when the internal virtual work is integrated by the midpoint rule and the external and inertial contributions by the trapezoidal rule. The proposed formulation inherits the robustness and the absence of locking from the underlying mixed FEM while simultaneously exposing a two-node coupling structure that mirrors discrete rod models from the computer graphics community. This is in sharp contrast to the dense coupling of strain-parameterized reduced-order models often used in soft robotic applications. Three numerical examples involving piecewise-varying cross sections, tendon-driven actuation under different spatial discretizations, and coupled longitudinal-torsional dynamics confirm the accuracy, robustness, and convergence behavior of the presented approach.

This work presents a complete framework for the modeling, workspace analysis, and trajectory planning of a tendon-driven continuum manipulator (TDCM). For modeling purposes, a Petrov–Galerkin Cosserat rod finite element formulation is adopted, and material parameters are identified through dedicated experiments. The identified model shows good agreement with the optical motion capture data, with a position error of no greater than 4.5

This paper presents a total Lagrangian mixed Petrov-Galerkin finite element formulation that provides a computationally efficient approach for analyzing Cosserat rods that is free of singularities and locking. To achieve a singularity-free orientation parametrization of the rod, the nodal kinematical unknowns are defined as the nodal centerline positions and unit quaternions. We apply Lagrange interpolation to all nodal kinematic coordinates, and in combination with a projection of non-unit quaternions, this leads to an interpolation with orthonormal cross-section-fixed bases. To eliminate locking effects such as shear locking, the variational Hellinger-Reissner principle is applied, resulting in a mixed approach with additional fields composed of resultant contact forces and moments. Since the mixed formulation contains the constitutive law in compliance form, it naturally incorporates constrained theories, such as the Kirchhoff-Love theory. This study specifically examines the influence of the additional internal force fields on the numerical performance, including locking mitigation and robustness. Using well-established benchmark examples, the method demonstrates enhanced computational robustness and efficiency, as evidenced by the reduction in required load steps and iterations when applying the standard Newton-Raphson method.

We propose a new, port-Hamiltonian formulation for the highly nonlinear dynamics of planar geometrically exact beams, which are amenable to arbitrary large deformations and rotations. A structure-preserving spatial and temporal discretization procedure - using mixed finite elements and second-order time-stepping methods - is proposed. It is observed that the present approach is objective, locking-free and provides an exact discrete representation of the energy and angular momentum balance. By comparing the approach to a classical displacement-based scheme from the literature it is shown that the port-Hamiltonian formulation paves new ways for the design of energy-momentum schemes in computational mechanics. Numerical examples underline the applicability to flexible multibody systems and beneficial numerical performance.

In this paper, we apply a numerical integration strategy recently developed for determining the deformation shapes of structures constituted by Cosserat rods, to predict the behavior of panto-cylinders. Panto-cylinders have, as microstructure, a set of two families of helicoidal beams interconnected by perfect or elastic joints. The pivot's free rotation axis is, in the reference configuration, orthogonal to the cylindrical surfaces spanned by the beams. We perform a series of numerical simulations looking for the mechanical parameters which exalt the chirality effects in the structure. For the performed compression, extensions, shear, and torsion tests, we find chiral deformation patterns with a dependence on the type of joint and its length.

In 1983, Andersen proposed the RATTLE algorithm as an extension of the SHAKE algorithm. The RATTLE algorithm is a well-established method for simulating mechanical systems with perfect bilateral constraints. This paper further extends RATTLE for simulating nonsmooth mechanical systems with frictional unilateral constraints (i.e. frictional contact). With that, it satisfies the need for higher-order integration methods within the framework of nonsmooth contact dynamics in phases where the contact status does not change (i.e. no collisions/constant sliding states). In particular, the proposed method can simulate impact-free motions, such as persistent frictional contact, with second-order accurate positions and velocities and prohibits penetration by unilateral constraints on position level.

Tendon-driven continuum soft robots are currently applied in research and are given a promising perspective for future applications. For the routing of the tendons from the actuator to the point where the loading is demanded, two routing possibilities exist in the literature: internal routing of the tendons with the help of structurally embedded Bowden sheaths and external tendon routing where the tendon is not in contact with the soft structure. The application of the latter is a clear disadvantage for applications due to the high risk of interference with the tendon, for example, causing the tendon to break. The first option on the other hand introduces high friction forces into the tendon transmission and affects the elastic characteristic of the continuum and therefore the desired workspace of the system. This article overcomes the aforementioned problems by integrating tendon routings within tendon channels eroded from the continuum structure by a model-based design method. The channels within the continuum structure are computed a priori such that the tendons do not interact with the continuum while moving through its workspace. Overall, a new model-based method for tendon channel design is introduced and a corresponding manufacturing process is established. A continuum joint module prototype is designed to enable roll-pitch-yaw motions with a large accessible workspace. The capabilities of the system are measured in experiments using an external camera for the range of motion. Moreover, walking experiments on the ANYmal robot from ETHZ are presented.

Based on more than three decades of rod finite element theory, this publication unifies all the successful contributions found in literature and eradicates the arising drawbacks like loss of objectivity, locking, path-dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jelenić in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SE(3)-structure of the Cosserat rod kinematics. Applying a Petrov-Galerkin projection method, we propose a novel rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time-derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parametrization in terms of a minimal number of nodal unknowns are demonstrated by conclusive numerical examples in both statics and dynamics.

A 2D-continuum model describing finite deformations in plane of discrete bi-pantographic fabrics has been recently obtained by applying an asymptotic procedure based on a set of local generalized coordinates. Rectangular bi-pantographic prototypes were additively manufactured by selective laser sintering using polyamide as raw material. Displacement-controlled bias extension tests were performed on such specimens for total elastic deformations up to ca. 25%. Experimental force measurements, complemented by discrete displacement measurements obtained by local digital image correlation, were used to fit the continuum model. In the present paper, a global and minimal set of generalized coordinates, alternative to the one used for the homogenization, is introduced for the discrete model. The mechanical constitutive parameters appearing in the discrete model are then found by means of collected experimental data. Finally, a comparison between experiments, the discrete and the continuum model is presented. It is concluded that (a) the discrete model and the experimental data are in excellent agreement, and that (b) the continuum retains the relevant phenomenology of the discrete system even for a rather low number of cells.

We present a continuum model for a three-dimensional pantographic structure that we call pantographic ortho-block. The model is based on the embedding of four families of elastic fibers in a three-dimensional domain. Similar to pantographic sheets, we find an energy density depending on the first and second-gradient of displacement resulting in a three-dimensional incomplete second-gradient material. For certain kinematic assumptions concerning perfect joints, we find two distinct zero-energy modes. Finally, we perform simulations based on isogeometric analysis to showcase standard and non-standard boundary conditions.

The application of the Petrov-Galerkin projection method in Cosserat rod finite element formulations offers significant advantages in simplifying the expressions within the discrete virtual work functionals. Moreover, it enables a straight-forward and systematic exchange of the ansatz functions, specifically for centerline positions and cross-section orientations. In this concise communication, we present a total Lagrangian finite element formulation for Cosserat rods that attempts to come up with the least required concepts. The chosen discretization preserves objectivity and allows for large displacements/rotations and for large strains. The orientation parametrization with non-unit quaternions results in a singularity-free formulation.

The standard in rod finite element formulations is the Bubnov-Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov-Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.

Dynamics of pantographic sheets presents exotic aspects that deserve investigation. In this paper, we focus the attention on some possible modalities of non-linear wave propagation in planar pantographic sheets. We use a smaller length-scale lattice model, in which the beams and pivots constituting the sheet are described by constrained Euler- Bernoulli beams together with a meso-reduced-order model which belongs to the class of second-gradient elastic materials and with a macro multi-field 1D continuum model, whose displacement is augmented by a specific class of cross-section deformations. Such a three-step reduction process is developed to allow for fast computational analysis of symmetric wave propagation patterns with respect to the longitudinal axis of the sheet. It is conceived by using suitable kinematical hypotheses for the 1D continuum descriptors referring to pantographic sheet sections which are inspired by the numerical evidence obtained performing simulations based on the smaller scale lattice model. The deformation energy of the pantographic sheet, successfully postulated in dell'Isola et al. (2016) for a meso-reduced-order second gradient model, is pivotal in the whole model reduction process: It allows for the determination of generalized 1D deformation energy in terms of the mechanical properties of the micro-lattice model. Performed numerical simulations prove that several waveforms propagate in planar pantographic sheets with low dispersion and motivate further investigations in the subject.(c) 2022 Elsevier B.V. All rights reserved.

Second-gradient continua are defined as those continua whose internal virtual work functionals depend on the first and second-gradient of the virtual displacement. These functionals can be represented either in Lagrangian (referential) or Eulerian (spatial) description thus defining respectively the Piola–Lagrange as well as the Cauchy–Euler stress and double-stress. In this paper, we deduce the Piola transformation formulae, i.e., those relationships between all Lagrangian and Eulerian fields relevant for the formulation of the Principle of Virtual Work. In particular, we derive the Piola transformations of stress and double-stress as well as the Piola transformations for external virtual work functionals compatible with second-gradient internal work functionals. The latter transformations contain in fact the Piola transformations of the contact surface and line forces as well as the contact surface double-forces.

In the theory of second-gradient continua, the internal virtual work functional can be considered as a second-order distribution in which the virtual displacements take the role of test functions. In its easiest representation, the internal virtual work functional is represented as a volume integral over a subset of the three-dimensional Euclidean vector space and involves first and second derivatives of the virtual displacements. In this paper, we show by an iterative integration by parts procedure how an alternative representation of such a functional can be obtained when the integration domain is a subset that contains also edges and wedges. Since this procedure strongly relies on the divergence theorem for submanifolds of a Euclidean vector space, it is a main goal to derive this divergence theorem for submanifolds starting from Stokes’ theorem for manifolds. To that end, results from Riemannian geometry are gathered and applied to the submanifold case.

Analysis of corrugated shell structures is an interesting problem in Structural Mechanics, which has many practical applications in Civil Engineering and Architecture. Thanks to corrugation, these structures have a remarkable feature: the wavy (undulated) shape in their edge provides significant enhancements in their structural behaviour, increasing the bending stiffness at the edge and allowing for a non-negligible reduction of its thickness. Moreover, looking at the non-linear behaviour, domes corrugation plays a relevant role in instability phenomena, such as the influence of imperfections and increasing resistance to snap-through. A problem in the study of such kind of shells is the definition of mathematical and geometrical model and the construction of a suitable mesh to perform FE analyses. The aim of this paper is to find an automated way to generate a double-curvature geometric surface that can be used both in static and in non-linear stability analyses of such corrugated shell structures. A method to generate a NURBS surface, suitable for a parametric FE analysis from a geometrical model expressed in a parametric form, is proposed and applied to a shell inspired by the well-known dome designed by Pier Luigi Nervi in 1959 for the roof of the Palasport Flaminio in Rome.